I will give a brief report on some the topics discussed at the workshop "Golod-Shafarevich groups and rank gradient" that took place this August in Vienna. I will focus on results involving rank gradient.

Saturated fusion systems are a relatively new class of objects that are often described as the correct 'axiomatisation' of certain p-local phenomena in algebraic topology. Despite these geometric beginnings however, their structure is sufficiently rigid to afford its own local theory which in some sense mimics the local theory of finite groups. In this talk, I will briefly motivate the definition of a saturated fusion system and discuss a remarkable result of Michael Aschbacher which proves that centralisers of normal subsystems of a saturated fusion system, F, exist and are themselves saturated. I will then attempt to justify his definition in the case where F is non-exotic by appealing to some classical group theoretic results. If time permits I will speculate about a topological characterisation of the centraliser as the set of homotopy fixed points of a certain action on the classifying space of F.

<p><span>I will present a history of the problem, relate it to other conjectures, and, with time permitting, indicate recent developments. The focus will primarily be group-theoretic and intended for the non-specialist.</span></p>